Question

Determine the Skolem forms of (a) \(\forall y \exists x\left(2 x^{2}+y x-1=0\right)\), (b) \(\forall \varepsilon \exists \delta(\varepsilon>0 \rightarrow(\delta>0 \wedge \forall x(|x-\bar{a}|<\delta \rightarrow|f(x)-f(\bar{a})|<\varepsilon)\), (c) \(\forall x \exists y(x=f(y))\), (d) \(\forall x y(x

Short answer

The Skolem forms are (a) \(2f(y)^2 + yf(y) - 1 = 0\); (b) \(\varepsilon > 0 \rightarrow (g(\varepsilon) > 0 \wedge \forall x (|x-\bar{a}|<g(\varepsilon) \rightarrow|f(x)-f(\bar{a})|<\varepsilon))\); (c) \(x = f(h(x))\); (d) \(x < y \rightarrow (u(x, y) < x) \wedge (y < v(x, y)) \wedge (x < v(x, y) \wedge w(x, y) < y)\); (e) \(x = f(x)^2 \vee x = -(f(x)^2)\).

Step-by-step solution

Understand General Skolemization

Skolemization involves removing existential quantifiers by introducing Skolem functions. For an existential quantifier \(\exists x\) following a universal quantifier \(\forall\), Skolemization replaces \(x\) with a Skolem function of the universally quantified variables before it. This results in a formula without existential quantifiers.

Skolemize Part (a)

Given the formula \(\forall y \exists x (2x^2 + yx - 1 = 0)\), Skolemization involves replacing \(x\) by a Skolem function of \(y\), say \(x = f(y)\). The Skolem form becomes \(2f(y)^2 + yf(y) - 1 = 0\).

Skolemize Part (b)

The formula is \(\forall \varepsilon \exists \delta (\varepsilon > 0 \rightarrow (\delta > 0 \wedge \forall x (|x-\bar{a}|<\delta \rightarrow|f(x)-f(\bar{a})|<\varepsilon)))\). Replace \(\delta\) by \(\delta = g(\varepsilon)\) to get \(\forall \varepsilon (\varepsilon > 0 \rightarrow (g(\varepsilon) > 0 \wedge \forall x (|x-\bar{a}|<g(\varepsilon) \rightarrow|f(x)-f(\bar{a})|<\varepsilon)))\).

Skolemize Part (c)

For \(\forall x \exists y (x = f(y))\), replace \(y\) by \(y = h(x)\) to obtain \(x = f(h(x))\) as the Skolem form.

Skolemize Part (d)

The formula is \(\forall x \forall y (x < y \rightarrow \exists u (u < x) \wedge \exists v (y < v) \wedge \exists w (x < v \wedge w < y))\). Introduce Skolem functions: replace \(u\), \(v\), and \(w\) by \(u = u(x, y)\), \(v = v(x, y)\), and \(w = w(x, y)\) respectively. So the Skolem form becomes \(x < y \rightarrow (u(x, y) < x) \wedge (y < v(x, y)) \wedge (x < v(x, y) \wedge w(x, y) < y)\).

Skolemize Part (e)

For \(\forall x \exists y (x = y^2 \vee x = -y^2)\), replace \(y\) by \(y = f(x)\), resulting in the Skolem form \(x = f(x)^2 \vee x = -(f(x)^2)\).

Key concepts

Skolem Function

The Skolem function is a central concept in logic, specifically in the process known as Skolemization. When dealing with predicate logic statements that contain existential quantifiers, the Skolem function allows us to remove these quantifiers by replacing them with specific function terms. This transformation helps simplify logical statements.

Here's how it works: when you encounter an existential quantifier of the form \( \exists x \), following a universal quantifier \( \forall y \), it is replaced with a function of \( y \), such as \( f(y) \). This function, called a Skolem function, is intended to capture all individual instances that \( x \) might represent, given the variable \( y \).

• For example, in the formula \( \forall y \exists x (2x^2 + yx - 1 = 0) \), the existential quantifier \( \exists x \) can be removed by introducing \( x = f(y) \). The formula then becomes \( 2f(y)^2 + yf(y) - 1 = 0 \).

Understanding Skolem functions allows for clearer representations of complex logical statements by eliminating existential control variables.

Existential Quantifiers

Existential quantifiers are symbols used in predicate logic to express that there exists at least one instance of something within a particular domain that satisfies a property. The symbol for an existential quantifier is \( \exists \). These quantifiers are crucial in statements where some element is guaranteed to exist but do not specify exactly what that element is.

For example, the statement \( \exists x (2x^2 + yx - 1 = 0) \) asserts that there is some value of \( x \) for which \( 2x^2 + yx - 1 = 0 \) holds true. However, when we apply Skolemization, we replace this existential quantifier with a Skolem function to find a precise representation of \( x \).

• This method is powerful as it converts a statement with an indeterminate existence into one with a defined function, making logical analysis more straightforward.

Understanding existential quantifiers is key to mastering the transition from ambiguous existence claims to clear logical expressions using Skolem functions.

Universal Quantifiers

Universal quantifiers are foundational components in predicate logic used to denote that a property holds for all elements within a given domain. The symbol used for universal quantifiers is \( \forall \). When you see a statement like \( \forall x P(x) \), it means that the property \( P \) applies to every instance of \( x \) in the domain.

In the context of Skolemization, universal quantifiers are often the "scope" within which an existential quantifier's range is adjusted. For example, if within \( \forall x \exists y \), \( y \) is dependent on each \( x \), through Skolemization, \( y \) becomes a Skolem function \( y = f(x) \).

• This transformation indicates the replacement of existential uncertainty with a function that consistently applies, based on \( x \).

Grasping universal quantifiers helps in understanding how generalized claims can be methodically addressed when converting existential quantifiers.

Predicate Logic

Predicate logic extends propositional logic by dealing with predicates, which are functions that can return true or false depending on the input. Unlike propositional logic, which considers statements such as "It is raining," predicate logic would consider "It is raining in location \( x \)." Here, \( x \) becomes a variable that can change the truth value of the predicate.

Predicate logic introduces quantifiers like \( \forall \) and \( \exists \), allowing for more sophisticated statements involving one or more variables. This leads to expressions where statements about each or some element of a domain can be made.

• The richness of predicate logic is evident in its ability to model real-world statements and logical operations.

In exercises where Skolemization is applied, predicate logic is the framework allowing for transformation through logical functions, simplifying the study of propositions involving existential and universal quantifiers.

Formal Logic

Formal logic is the systematic study of inference with mathematical rigor. It focuses on defining precise symbols and rules for analyzing arguments. The primary goal is to develop systems where logical forms can be manipulated without the ambiguity found in natural language.

Within formal logic, predicate calculus takes a critical role. With predicates, quantifiers, and logical connectors, formal logic aids in framing, solving, and interpreting logical statements like those encountered in Skolemization.

• Formal logic helps transform statements with existential and universal quantified expressions into more manageable forms.

In the structured environment of formal logic, Skolemization serves as a tool to substitute existential quantifiers using specific functions, ensuring clarity and precision in logical procedures. Mastery of formal logic enables profound engagement with complex arguments by systematically removing uncertainties inherent in existential claims.